A WEIGHTED ESTIMATE OF COMMUTATORS OF BOCHNER–RIESZ OPERATORS FOR HERMITE OPERATOR

Abstract

Let H be the Hermite operator $-\Delta +|x|^2$ on $\mathbb {R}^n$. We prove a weighted $L^2$ estimate of the maximal commutator operator $\sup _{R>0}|[b, S_R^\lambda (H)](f)|$, where $ [b, S_R^\lambda (H)](f) = bS_R^\lambda (H) f - S_R^\lambda (H)(bf) $ is the commutator of a BMO function b and the Bochner–Riesz means $S_R^\lambda (H)$ for the Hermite operator H. As an application, we obtain the almost everywhere convergence of $[b, S_R^\lambda (H)](f)$ for large $\lambda $ and $f\in L^p(\mathbb {R}^n)$.